Buildings, Bruhat decompositions, unramified principal series

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1 (July 2, 2005) Buildings, Bruhat decompositions, unramified principal series Paul Garrett garrett/ [ ] [ ] [ ] [ ] EDIT:... draft... Buildings are special simplicial complexes on which interesting groups such as GL n (Q) and SL n (Q p ) act in an illuminating fashion. The geometric structure is critical, and the geometric language is a powerful heuristic and mnemonic. This viewpoint was created by Jacques Tits as a way to subsume the geometric algebra that is nearly sufficient to treat basic properties of classical groups such as general linear groups, symplectic groups, orthogonal groups, unitary groups, while giving a stronger language and viewpoint allowing treatment of the exceptional groups. Further, Bruhat and Tits subsequently gave an intrinsic treatment of buildings attached to reductive groups. Here we will give an economical treatment that presumes few prerequisites, but hopefully does not degrade the central ideas. [ ] The first application is to Bruhat decompositions, whose first assertion is a decomposition into cells [ ] G = w W In the simplest case where G = GL n (k) (invertible n-by-n matrices with entries in a field k), the minimal parabolic [ ] P can be taken to be upper-triangular (invertible) matrices, and the Weyl group [ ] W can be taken to be all permutation matrices. It is true that it is possible to give a direct ad hoc proof of this fact for GL n (k), for example. However, the ad hoc argument is arduous and unilluminating, and, for example, does not easily give the disjointness of the union, for larger n. Further, refinements of the decomposition for non-minimal parabolic subgroups, are less accessible by seemingly elementary methods. As a global application, refinements of the Bruhat decomposition are essential to the discussion of constant terms of Eisenstein series, and in understanding their meromorphic continuations. Such issues lie behind both Langlands-Shahidi and Rankin-Selberg integral representations of L-functions. In the mid-1960s Iwahori and Matsumoto made the surprising discovery that the Bruhat-Tits buildings formalism, conceived as a device to study parabolic subgroups, was applicable to a very different issue, that of compact open subgroups in p-adic reductive groups, such as SL n (Q p ). In particular, this brought to light the technical importance of some subtler items than considered immediately from classical motivations. For example, rather than maximal compact open subgroups as fundamental objects, somewhat smaller compact open subgroups (now called Iwahori subgroups) were shown to be the fundamental gadgets. An application of critical importance for the representation theory of p-adic groups stemming from the Iwahori-Matsumoto discovery is the Borel-Matsumoto theorem (circa 1976) that asserts (among other things) that an irreducible representation of a p-adic group possessing an Iwahori-fixed vector has an imbedding into It turns out that it is possible to ignore almost completely the traditional combinatorial group theory usually taken as a prerequisite. That is, we find that the theory of Coxeter groups is not necessary for what we do here. Under various further hypotheses, the sets P wp are literal cells in the topological sense of being homeomorphic to open balls. We will not need such particular details, but nevertheless will refer to the sets P wp as Bruhat cells. We give a building-theoretic definition below. There is also a definition coming from the viewpoint of algebraic geometry. The latter is not immediately helpful to our present purposes, and even the comparison to the buildingtheoretic definition does not help us, so we neglect the definition from algebraic geometry. There are many forms of a general definition of Weyl group. We give a building-theoretic definition below. Comparison with other definitions is a job in itself, which we do not undertake. P wp 1

2 an unramified principal series representation. [ ] Further, intertwining operators among unramified principal series can be understood well enough from the Borel-Matsumoto viewpoint (Casselman 1980) so as to give very clear criteria for irreducibility of unramified principal series and even degenerate principal series [ ] The theme of groups acting on things is pervasive. Further, often as much interest resides in the proof technique as in the results themselves. Even in the simplest example, the Sylow theorems, the fact that the action of the group on p-subgroups by conjugation is productive is more interesting than the specific conclusions of the theorem. In the case of buildings and groups acting on them the things on which the groups act are now more structured, and more subtly structured, than in simpler examples. Simplicial complexes, chamber complexes, buildings Example: spherical building for GL(n) Canonical retractions, uniqueness lemma Group actions, parabolic subgroups Weyl groups, Bruhat decompositions Reflections, foldings Bruhat cell multiplication EDIT:... more later Simplicial complexes, chamber complexes, buildings This section introduces geometric language necessary to talk about buildings. A simplex is a generic member of the family of geometric objects including points, line segments, triangles, tetrahedrons, and so on. A 0-simplex is a point, a 1-simplex is a line segment, a 2-simplex is a triangle, a 3-simplex is a solid tetrahedron. fig 1 ideal 0-simplex ideal 1-simplex ideal 2-simplex ideal 3-simplex But for present purposes, we need only a shadow of geometry. The idea is that an n-simplex should be determined by its n + 1 vertices, so a simplex is a set (of vertices). [ ] [ ] We will define and illustrate these representations later. representations of p-adic groups. They are the most accessible and understandable of [ ] By definition, these are induced from one-dimensional characters on non-minimal parabolics. They are proper subrepresentations of unramified principal series, so it is not immediately clear how study of reducibility of unramified principal series helps us understand irreducibility of subrepresentations which only occur in reducible unramified principal series. More on this below. [ ] Thus, two simplices with the same set of vertices are identical. 2

3 fig 2 our 0-simplex our 1-simplex our 2-simplex our 3-simplex The faces τ of a simplex σ are the simplices appearing as non-empty subsets τ σ. [ ] a simplex σ is one less than the cardinality of the underlying set, [ ] namely The dimension of dim σ = σ 1 The codimension of a face τ of a simplex σ is the difference of dimensions codimension of τ in σ = dim σ dim τ fig 3 two faces of 1-simplex 3 codim-one faces of 2-simplex 4 codim-one faces of 3-simplex A simplicial complex X is a set V of vertices, and a distinguished set X of subsets of the vertices, with the property that σ C and τ σ implies τ C. The sets of vertices [ ] in X are the simplices in the complex, and that last requirement is that if a simplex σ is in X then all the faces of σ are in X as well. The dimension of a simplicial complex is the maximum of the dimensions of the simplices in it. A subcomplex Y of a simplicial complex X is a simplicial complex which has vertices which form a subset of the vertices of X, and has simplices which are a subset of those of X. In a simplicial complex, two simplices of the same dimension are adjacent if they have a common codimension-one face. fig 4 simplicial complex two adjacent simplices shaded [ ] [ ] [ ] Thus, the intersection of two simplices is a common face of the two. In this discretized geometry, 1-simplices (line segments) can only intersect at endpoints (or not at all), and two 1-simplices with both endpoints in common are identical. 2-simplices (triangles) can only intersect in vertices, A point is zero-dimensional, a one-dimensional line segment needs 2 points to specify it, a two-dimensional triangle needs 3 points, and so on. The 0-dimensional faces τ of a simplex σ are the singleton subsets {x} for x σ. That is, the 0-dimensional faces are {x} for vertices x of σ. The distinction between x and the singleton set {x} is usually not important. 3

4 A simplicial complex map f : X Y is a set-map on the underlying sets of vertices, with the property that f is a bijection on every simplex in X, and for a simplex σ in X, the image f(σ) is a simplex in Y. [ ] A chamber in a simplicial complex X is a simplex which is maximal, in the sense of not being a face of any other simplex in X. A gallery C 0,..., C n connecting two chambers C 0 and C n in a simplicial complex X is a sequence of chambers C i such that C i and C i+1 are adjacent. The length of that gallery is n. The distance between two chambers is the length of a shortest gallery connecting them. [ ] A chamber complex is a simplicial complex in which any two chambers are connected by a gallery. [ ] A chamber complex is thin if every codimension-one face is a face of exactly two chambers. [ ] shading shows fig 5 C D gallery connecting chambers C and D in thin chamber complex A chamber complex is thick if every codimension-one face is a face of at least three chambers. [ ] fig 6 a one-dimensional thick building (infinite) homogeneous tree, chambers are line segments A simplicial subcomplex Y of a chamber complex X is a chamber subcomplex if it is a chamber complex with chambers of the same dimension as those in X. A chamber-complex map is a simplicial complex map on chamber complexes of the same dimension. [ ] Thus, a simplicial complex map f : X Y need not be either injective or surjective on the sets of vertices of X and Y, but respects simplices in X in the sense that it neither reduces their dimension nor tears them apart (by mapping them to non-simplices in Y ). We can also say that a simplicial complex map preserves face relations, since by its behavior on vertices, if τ is a face of σ in X, then f(τ) is a face of f(σ) in Y. [ ] [ ] [ ] [ ] Thus, two adjacent chambers are at distance 1. This implicitly requires that any two chambers are of the same dimension. Thin chamber complexes are vaguely like manifolds, or manifolds with boundary. Vaguely, thick chamber complexes are like bunches of manifolds stuck together at various patches. 4

5 A simplicial complex X is a (thick) building if X is a thick chamber complex. There is a set of chamber subcomplexes (the apartments) of X such that Each apartment is a thin chamber complex. Any two chambers in X are contained in a common apartment. For two apartments a, a containing chambers C and D in common, there is a chamber-complex isomorphism f : a a fixing the vertices of both C and D. [ ] Remark: Note that the chamber-complex property of the whole building X will follow from the two fact that the apartments are chamber complexes, and from the fact that any two chambers are contained in a common apartment. All that needs to be proven separately about the building is the thickness. 2. Example: spherical building for GL(n) Here we construct a family of buildings on which the k-linear automorphisms of a vector space V over a field k will act nicely. This construction will be used to study the group GL(n). [ ] Specifically, groups such as the subgroup of upper-triangular matrices in GL(n) will arise, by design, as stabilizers of chambers. We construct the simplicial complex, then prove that it meets the requirements for a thick building. There is non-trivial substance in the argument required to verify that this is a building. Let V be an n-dimensional vectorspace over a field k. A flag in V is a nested chain V d1... V dt with proper inclusions, of vector subspaces [ ] of V. Often the subscript denotes the dimension. The length of the flag is the number of subspaces in it. A flag is maximal if it cannot be made longer. Let X be the simplicial complex whose vertices are proper vector subspaces of V, and whose maximal simplices are maximal flags C of proper subspaces of V [ ] [ ] V 1 V 2... V n 1 where V i is of dimension i. The faces of this simplex C are the (non-empty) subflags F of σ, namely all (non-empty) flags F V i1... V il (necessarily i 1 <... < i l ) The subcomplexes we propose as apartments consists of subcomplexes a specified by frames, that is, by unordered collections of n one-dimensional vector subspaces L 1,..., L n which span the vectorspace. A frame In fact, each apartment in a building has the structure of a Coxeter complex, meaning the following. First, a Coxeter group is a group G with generators S and relations as follows. For all s S, s 2 = e. For every s, t S there is m = m(s, t) (possibly infinite, meaning no relation) such that (st) m = e. There are no other relations. The generalized reflections in G are the conjugates of elements of S. Two group elements g, h are adjacent if there is a generalized reflection t in G such that tg = h. Beginning with this idea, one can construct a thin chamber complex, the associated Coxeter complex from a Coxeter group. The usual way of defining a building includes a requirement that apartments be Coxeter complexes. However, such a definition has disadvantages, and we proceed differently. There are several related but mutually incompatible notations. First, for positive integer n and field k, GL n(k) is the group of invertible n-by-n matrices with entries in k. This is sometimes denoted by GL(n, k), or even GL(n) when the reference to k is implicit or irrelevant. So far, this is fairly self-consistent. However, for a k-vectorspace V (without a choice of basis), one may also write GL k (V ) for the group of k-linear automorphisms of V. [ ] For our purposes we prohibit the 0-subspace and the whole space in flags. 5

6 Paul Garrett: Buildings (July 2, 2005) determines a subcomplex a by including in a all flags involving only subspaces expressible as sums of lines from the frame. fig 7 vertices of two types: lines and planes vertices of three types extreme vertices are lines edge midpoints are planes face centers are 3-folds apartment for GL(3) apartment for GL(4) (barycentric subdivision of tetrahedron) Claim: The simplicial complex X of flags in V, with apartments determined by frames, is a thick building. Proof: We must verify four things, that the whole is thick, [ ] that each apartment is a thin [ ] chamber complex, [ ] that any two simplices are contained in a common apartment, and, last, that for two apartments a and b containing common chambers C, D there is a chamber-complex isomorphism ϕ : a b fixing the vertices of both C and D. Let C be a top-dimensional simplex in X, namely a maximal flag V 1... V n 1 where V i is i-dimensional. The codimension-one faces of C are the simplices τ i given by omitting the i th subspace, that is, V 1... V i 1 V i+1... V n 1 Consider the case that the index i is properly between the end values, that is, 1 < i < n. Then the other maximal simplices with τ i as face are obtained by replacing V i by some other i-dimensional subspace lying between V i 1 and V i+1. The collection of such subspaces is in bijection with the set of lines the quotient V i+1 /V i 1. This quotient is a two-dimensional k-vectorspace. If k is infinite, there are infinitely-many such lines. If k is finite, with q elements, then there are (q 2 1)/(q 1) = q + 1 lines, which is at least 3. This proves the thickness. [ ] The cases i = 1 and i = n are nearly identical, except for notation. To prove that the apartments are thin chamber complexes, we need to prove that they are chamber complexes in the first place, and that they are thin. Fix a frame F. As in the discussion of the last paragraph, for a maximal simplex C in X given by a maximal flag V 1... V n 1 the other maximal simplices adjacent along the i th face V 1... V i 1 V i+1... V n 1 are obtained by replacing V i by another i-dimensional subspace lying between V i 1 and V i+1. In a fixed apartment, the choice of this i-dimensional subspace is constrained. In particular, for some pair of (distinct) lines L 1, L 2 in the frame, V i+1 = V i 1 L 1 L 2 [ ] [ ] [ ] Again, thickness is that each codimension-one face is the face of at least three chambers. Again, thin-ness is that each codimension-one face is the face of exactly two chambers. Again, a chamber complex is a simplicial complex in which any two maximal simplices are connected by a gallery. [ ] As noted earlier, since we will prove that in apartments any two chambers are connected by a gallery, and we will prove that any two chambers lie in a common apartment, we will have proven that in the building as a whole any two chambers are connected by a gallery. That is, the whole building is a chamber complex. 6

7 Thus, there are just two choices of i-dimensional subspace between between V i 1 and V i+1, namely This proves the thin-ness of apartments. V i 1 L 1 and V i 1 L 2 Now prove that any two maximal simplices in an apartment are connected by a gallery. Let C be the chamber given by the maximal flag V 1... V n 1 where V i = L 1... L i with lines L 1,..., L n specifying the frame. Thus, choice of a chamber is equivalent to choice of an ordering of the lines in the frame. From the previous paragraph, the chamber adjacent to C across the i th codimensionone face V 1... V i... V n 1 is V 1... V i 1 V i 1 L i+1 V i+1... V n 1 That is, moving across the i th face interchanges the i th and (i 1) th lines in the ordering specifying the chamber. Since every permutation of n things is expressible as a product of adjacent transpositions, [ ] every ordering of lines is obtained in such fashion. That is, any two chambers in the apartment are connected by a gallery. Now prove that any two chambers C, D are contained in a common apartment. That is, we find a frame in terms of which all the subspaces occurring in C or in D are expressible. [ ] Let C be the maximal flag V 1... V n 1 and let D be the maximal flag W 1... W n 1 In order to find a common frame for the two flags, we recall some standard but slightly technical points. Lemma: For subspaces X, Y, η of a k-vectorspace V, with Y η, (X + η) Y = (X Y ) + η Proof: The proof is inevitable, once one realizes that this is what we ll need. (See proof of Zassenhaus theorem just following.) First, (X Y ) + η Y (X + η) where or not η Y. On the other hand, let x + y = y with x X, y Y, and y η. Then x = y y X (Y η) = X Y from which y is in (X Y ) + η. [ ] [ ] This is standard, proven by induction on n: Let π be a permutation of n things, that is, a bijection of {1,..., n} to itself. If π(n) = n, then π can be identified with a permutation of n 1 things, and we re done, by induction. For π(n) = m < n, do induction on n m. Let s be the permutation which interchanges m and m + 1 and does not move any other element. Then (s π)(n) = m + 1, and induction finishes the argument. Finding the frame is an application of Zassenhaus theorem, which is an idea preliminary to the proof of Jordan- Hölder-type theorems. 7

8 [ ] Paul Garrett: Buildings (July 2, 2005) In a slightly different notational style, possible since we ll not need to refer to elements: Theorem: (Zassenhaus) [ ] Let X x and Y y be subspaces of a vector space V. Then there are natural isomorphisms (X + y) Y (x + y) Y X Y (x Y ) + (X y) (x + Y ) X (x + y) X Proof: The kernel of is Applying the previous lemma twice gives X Y (X + y) Y (X + y) Y (x + y) Y (X Y ) ((x + y) Y ) = X (x + y) Y X (x + y) Y = X ((x Y ) + y) = (X η) + (x Y ) This gives the left isomorphism. The right isomorphism follows by reversing the roles of X, x and Y, y. Now we return to the proof that there is a common apartment for two given chambers. First, as a matter of notation, let V n = W n = V. For a given index i, dim k V i /V i 1 = 1, so there is a smallest index j such that Then dim k so (V i W j 1 ) + V i 1 = V i 1, and, thus, For l > j, still but also so V i V i 1 = dim k (V i W j ) + V i 1 V i 1 = 1 dim k (V i W j 1 ) + V i 1 V i 1 = 0 V i V i 1 (V i W j ) + V i 1 V i 1 (V i W j ) + V i 1 (V i W j 1 ) + V i 1 dim k (V i W l ) + V i 1 V i 1 = 1 dim k (V i W l 1 ) + V i 1 V i 1 = 1 dim k That is, given i, there is a unique index j such that (V i W l ) + V i 1 (V i W l 1 ) + V i 1 = 0 V i V i 1 (V i W j ) + V i 1 V i 1 (V i W j ) + V i 1 (V i W j 1 ) + V i 1 Then, via Zassenhaus theorem, given i, there is exactly one index j such that V i V i 1 (V i W j ) + V i 1 (V i W j 1 ) + V i 1 V i W j (V i W j 1 ) + (V i 1 W j ) This result is sometimes called the Butterfly Lemma, due to the fact that one can manage to draw a diagram indicating the relations among the various subspaces that resembles a butterfly. 8

9 Invoking Zassenhaus theorem again, Paul Garrett: Buildings (July 2, 2005) V i W j (V i W j 1 ) + (V i 1 W j ) (V i W j ) + W j 1 (V i 1 W j ) + W j 1 By a symmetrical argument, for given j, the right-hand side of the last isomorphism is one-dimensional for exactly one index i. Thus, we have shown that there is a bijection i j of {1,..., n} to itself such that all these quotients are one-dimensional. Given a pair i and j, let L i be a one-dimensional subspace of V mapping surjectively to both V i /V i 1 and W j /W j 1. Then sums of the lines L 1,..., L n express all subspaces V i and W j, so give a frame specifying an apartment in which both the given chambers lie. That is, we have proven that there is an apartment containing any two given chambers. Last, we verify that for chambers C, D lying in the intersection a b of two apartments, there is a simplicial complex isomorphism f : a b fixing C and D pointwise. In fact, letting L 1,..., L n and M 1,..., M n be the one-dimensional subspaces specifying the two apartments, we will give a bijection between these sets of lines which will yield the identity map on the two given chambers. By renumbering the lines if necessary, we can suppose that the chamber C corresponds to the orderings L 1,..., L n and M 1,..., M n, that is, to the flags L 1 L 1 L 2... L 1... L n 1 Define a map on vertices by M 1 M 1 M 2... M 1... M n 1 f : a b f(l i1... L im ) = M i1... M im ) for any m-tuple of indices i 1 <... < i m. On the chamber C this is the identity. By the uniqueness lemma, if there is an isomorphism a b, this map must be it. [ ] It suffices to prove that f is the identity map on a b, and to prove this it suffices to prove that f is the identity on vertices in that intersection. We claim that L i1... L im = M j1... M jm with i 1 <... < i m and j 1 <... < j m implies that i l = j l for all l. We prove this by induction on m. The case m = 1 is trivial. For m > 1, let l be the largest [ ] index such that i l j l. Without loss of generality, suppose that i l < j l. Our hypothesis about the numbers of the lines and the expressibility of C in terms of both gives L 1 L 2... L jl 2 L jl 1 = M 1 M 2... M jl 2 M jl 1 Adding these subspaces to the given one yields L 1 L 2... L jl 2 L jl 1 L il+1... L im = M 1 M 2... M jl 2 M jl 1 M jl... M jm For all µ > m we have i µ = j µ. Thus, taking dimensions, since i l < j l, (j l 1) + (m l) = (j l 1) + (m l + 1) which is impossible. Thus, i l = j l for all l, and f : a b is an isomorphism. This proves that our construction gives a building. [ ] The fact that the second chamber D played no role in the definition of f is less surprising by this point, in view of the uniqueness lemma. [ ] Yes, largest, not smallest. 9

10 Paul Garrett: Buildings (July 2, 2005) Remark: The group GL k (V ) of k-linear automorphisms of the vectorspace V acts by simplicial-complex maps on the building X just constructed, since GL k (V ) preserves dimension and containment of subspaces. But we do not yet have sufficient information to do much with this yet. Remark: These buildings are spherical, since, with some trouble, one can verify that the apartments are simplicial versions of spheres. We will make no use of this, so will not worry about justifying the terminology. 3. Canonical retractions, uniqueness lemma This section contains the first non-trivial abstract results on buildings. A retraction [ ] r : X Y of a simplicial complex X to a subcomplex Y of X is a simplicial complex map whose restriction to Y is the identity map. Two simplicial complex maps agree pointwise if they are equal on vertices, hence on all simplices and their faces. A gallery C 1,..., C n in a chamber complex stutters if a chamber appears twice or more consecutively, that is, if for some index C i = C i+1. Theorem: Given an apartment a in a building X, there is retraction X a. Indeed, given a chamber C in a, there is a unique retraction X a sending non-stuttering galleries starting at C to non-stuttering galleries in a (necessarily starting at C). Further, this retraction is an isomorphism a a on any apartment a containing C. This retraction is the canonical retraction of the building to the given apartment, centered at the given chamber. Proof: As a first step toward construction retractions, we prove a result important in its own right. Lemma: (Uniqueness) Let X, Y be chamber complexes, with Y having the property [ ] that each codimension-one face is a face of at most two chambers. Let r : X Y, g : X Y be chamber complex maps which agree pointwise [ ] on a chamber C in X, and both f and g send non-stuttering galleries starting at C to non-stuttering galleries. Then f = g. Proof: (of lemma) Let C = C 0, C 1,..., C n = D be a non-stuttering gallery. By hypothesis, its image under f and its image under g do not stutter. That is, fc i fc i+1 for all i, and similarly for g. Suppose, inductively, that f agrees with g on C i and all its faces. Certainly fc i and fc i+1 are adjacent along the face F = fc i fc i+1 = gc i gc i+1 By the non-stuttering assumption, fc i+1 fc i and gc i+1 gc i. Thus, by the hypothesis on Y, it must be that fc i+1 = gc i+1, since there is no third chamber with facet F. Since there is a gallery from C to any other chamber, this proves that f = g pointwise on all of X. Remark: The uniqueness lemma allows formulation of a more memorable version of one of the defining conditions for a building. That is, rather than the original requirement that for any two apartments containing a chamber C and a simplex σ there is a simplicial complex isomorphism f : a a fixing C and σ pointwise, we have the following. [ ] [ ] [ ] This notion of retraction is a discretized version of the usual notion in topology. The hypothesis on Y is certainly met for Y thin, but we need the slightly weaker hypothesis later. Again, pointwise agreement means agreement on vertices. 10

11 Paul Garrett: Buildings (July 2, 2005) Corollary: For two apartments a, a A containing a common chamber C, there is be a chamber-complex isomorphism f : a a fixing a a pointwise. Proof: This implies the original axiom. For a simplex σ a a, there is an isomorphism f σ : a a fixing σ and C pointwise, by the building axiom. The Uniqueness Lemma implies that there can be at most one such map which fixes C pointwise. Thus, f σ = f τ for all simplices σ, τ in the intersection. Now we try to construct a retraction r : X a of X to a. For a chamber D not in a, let a be an apartment containing both C and D, and f : a a an isomorphism which pointwise fixes a a. The existence of f is assured by the last corollary. By the uniqueness lemma, there is just one such f. For another apartment a containing both C and D, let f : a a be the unique isomorphism which fixes a a pointwise. We claim that f D = f D, so that we can define rd = f D = f D Let g : a a be the isomorphism fixing D pointwise, from the building axioms. Then by uniqueness f g = f, that is, the diagram a g a f f a commutes. Then on a a the map f g is f. That is, these isomorphisms to a agree on overlaps, so give a well-defined retraction r to a. Corollary: Let C and D be two chambers in X. Let a be an apartment containing both C and D. Then the length of a shortest gallery from C to D inside a is the same as the length of a shortest gallery from C to D inside the whole building X. Proof: Let r be the retraction of X to a centered at C. Then the image under r of a gallery from C to D in X is no longer than the original gallery. 4. Group actions, parabolic subgroups We want simplices in buildings to have no non-trivial automorphisms, so that fixing a simplex will mean fixing it pointwise. To achieve this, we want to distinguish types of vertices, rather than seeing all vertices as the same. [ ] Specifically, a typing or labeling of an n-dimensional chamber complex X is a simplicial complex map [ ] λ : X where is a simplex. [ ] The type or label of a vertex is its image by λ in. Given a labeling [ ] X, a simplicial complex map f : X X of X to itself is label-preserving [ ] A similar finer distinction is necessary in algebraic topology, where a notion of orientation of a simplex is introduced. This amounts to ordering the vertices, modulo even permutations. The initial confusion about this parity distinction in some cases motivated treatment of homology modulo 2, for no better reason than to avoid worry about signs. [ ] Recall that this means that dimensions of simplices are preserved, and implies that face relations are preserved. [ ] [ ] Necessarily is of dimension at least that of X. We will only care about the case that has the same dimension as X. In fact, it can be shown, with some effort, that every thick building has a labeling, but all our constructions will make a concrete labeling evident. Thus, we need not dally to prove the general fact, which would entail that we prove that each apartment is a Coxeter complex, etc. 11

12 Paul Garrett: Buildings (July 2, 2005) or type-preserving if λ f = λ, that is, if we have a commuting diagram X f X λ λ Example: In the spherical building X for an n-dimensional vectorspace V over a field k, the vertices are linear subspaces of V, and can be labeled by their dimension. That is, let = {1, 2,..., n 1} and λ : X by λ(x) = dim k x for vertices x of X. The natural action of G = GL k (V ) on subspaces certainly preserves dimension, so the action of GL k (V ) is label-preserving. Let X be a thick building with labeling λ : X. Let G be a group acting on X [ ] by simplicial complex maps preserving labels. The group action is said to be strongly transitive if it is transitive on pairs C, a, where C is a chamber in an apartment a. [ ] A parabolic subgroup P in G is a stabilizer of some simplex σ in the building, that is, P = {g G : gσ = σ} The minimal parabolics are stabilizers of chambers. Maximal parabolics are stabilizers of vertices. [ ] Example: In the action of GL(n) on the (n 1)-dimensional spherical building attached to an n-dimensional vectorspace V over a field k, the parabolic subgroups admit visually memorable descriptions in terms of matrices. Let e 1,..., e n be the standard basis for V = k n, and identify G = GL k (V ) with GL n (k), the group of n-by-n invertible matrices with entries in k. Take chamber C specified as maximal flag ke 1 ke 1 ke 2... ke 1... ke n 1 The minimal parabolic stabilizing this flag is the so-called standard minimal parabolic consisting of uppertriangular matrices The n 1 different maximal parabolics fixing faces of C are the fixers of length-one flags ke 1... ke i and consist of matrices where the two diagonal blocks are invertible. The general parabolic fixing some face of C has square blocks of varying sizes along the diagonal, zeros below, and anything above. [ ] [ ] [ ] A building always needs an implied system of apartments. We will not worry about possibly varying choices of such. Implicit in this is that the group maps apartments to apartments. As usual, smaller sets are stabilized by larger subgroups, and vice-versa. 12

13 fig 8 [ white areas are 0 shaded areas non-trivial ] maximal parabolic another maximal parabolic their intersection: next-to-maximal parabolic Transitivity gives some easy but important results that arise in all group actions on sets: Proposition: All minimal parabolic subgroups are conjugate in G. Proof: This is essentially because the group is postulated to act transitively on chambers. In more detail, as usual, let P be the stabilizer of a chamber C and Q the stabilizer of a chamber D. Let g G be such that gc = D. For q Q, q(gc) = qd = D = gc so apply g 1 to obtain (g 1 qg)c = C so g 1 qg P. That is, g 1 Qg P. The argument is clearly reversible, so we have equality. Remark: Because of the labeling, with an n-dimensional building there are n + 1 conjugacy classes of maximal parabolics, since G preserves labels. Remark: For GL(n), from our present viewpoint the causality will run the other way, that is, we will prove the strong transitivity of the group by looking at the explicit behavior of flags of subspaces. Example: We claim that the natural action of GL k (V ) on the spherical building constructed earlier is strongly transitive. Label the vertices of the spherical building by dimension of the subspace (which is the vertex). The natural action of GL k (V ) on linear subspaces of V certainly preserves dimensions of subspaces, so preserves labels. Apartments are specified by frames, that is, unordered collections {L 1,..., L n of lines (one-dimensional subspaces) L i whose direct sum is the whole space V. As in the earlier proof that this complex truly is a building, the chambers within the apartment specified by a frame F are in bijection with the orderings of the lines L i. To prove strong transitivity is to prove that GL k (V ) is transitive on sets of one-dimensional subspaces whose direct sums are the whole space V. Indeed, for another set µ 1,..., µ n, there is a unique invertible k-linear map which sends L i µ i for all i. This proves the strong transitivity in this example. [ ] 5. Weyl groups, Bruhat decomposition There are more subgroups of interest that can be nicely specified in terms of the action on the building. Again, let X be a thick building with a labeling λ : X, and let G be a group acting on X strongly transitively, preserving labels. Let N = N (a) = stabilizer of an apartment a A = A(a) = pointwise fixer of an apartment a W = W (a) = N /A = Weyl group of an apartment a [ ] In this example, the strong transitivity has little content. By contrast, the proof that the alleged building is indeed a building is more serious. 13

14 Proposition: The Weyl group W of an apartment a has a well-defined action on a by invertible simplicial maps. The group W acts transitively on the chambers in the apartment, and is in bijection with the chambers by w wc for any fixed chamber C in a. Proof: The strong transitivity immediately shows that N stabilizes a and is transitive on chambers in a. Since by definition A fixes a pointwise, the quotient W = N /A has a well-defined action on a. Let w W fix a given chamber C. The label-preserving property of the group action implies that w fixes C pointwise. Since the elements of W give isomorphisms on a, they certainly send non-stuttering galleries to non-stuttering galleries. Thus, by the uniqueness lemma, there is at most one isomorphism w : a a to the thin chamber complex a fixing C. The identity map on a is one such, so w is necessarily the identity on a. Thus, the map w wc must be a bijection. For a fixed chamber C in the apartment a in X, the corresponding minimal parabolic P is The following is of central importance. [ ] P = minimal parabolic = stabilizer of C Theorem: (Bruhat decomposition) There is a disjoint decomposition G = w W In more detail: let r : X a be the retraction to a centered at C. Given g G let w W be the unique element such that wc = r(gc). Then g P wp P wp Remark: Note that although W = N /A is not a subgroup of G, since A P any double coset P wp for w W is well-defined. And recall that W is in bijection with the chambers in a, from above. Proof: By the building axioms, there is an apartment a containing both gc and C. The retraction r restricted to a is an isomorphism to a, fixing C pointwise. Likewise, the strong transitivity of G on X implies that there is an element p P mapping a to a. By uniqueness, these two maps a a must be identical. That is, pgc = r(gc) = wc That is, (w 1 pg)c = C so w 1 pg P. Remark: In fact, we could have done without the retraction r : X a entirely, but its existence exhibits the coherence among the various isomorphisms of other apartments to a given one. The form of the Bruhat decomposition above suffices for many applications, but not all. For example, for the Borel-Matsumoto theorem on unramified principal series, we will need finer information on the Weyl group and on Bruhat-like decompositions. We begin the clarification of the nature of the Weyl group by the following discussion. [ ] Even in very simple situations, where instances of this result admit easy proofs, the fact seems to have only been discovered in the 1950s. 14

15 Proposition: Fix a chamber C in an apartment a in X. For each chamber D in a adjacent to C there is a unique element s W (the reflection along C D) such that sc = D and sd = C. This reflection s has the property that s 2 = 1. The collection S of all reflections along codimension-one faces of C generates W. Proof: First, we prove existence of a reflection along each codimension-one face of C. Given adjacent chambers C, D in a, by strong transitivity of the action of G there is an element s G stabilizing a and sending C to D. Since s stabilizes a it lies in N, and if we restrict our attention to the effect of s on a we may as well view s as lying in the quotient W = N /A. Since G is label-preserving, s must fix pointwise C D. Thus, the single vertex of C not lying in C D must be mapped by s to the single vertex of D not lying in C D, and vice-versa. This proves existence. Since s : a a is an isomorphism, it maps non-stuttering galleries to non-stuttering galleries, so by the uniqueness lemma the effect of s on a is completely determined by the fact that it maps C D fixing C D pointwise. This proves uniqueness of the reflection along C D. Since s 2 is an automorphism of a fixing C pointwise, s 2 is the identity map on a, again by the uniqueness lemma. To prove that the set S of reflections along codimension-one faces of C generates W, do induction on the length of a minimal gallery from C to a chamber wc for w W. Let C, sc,..., wc be a minimal gallery for some s S. Apply s to this gallery to obtain sc, C,..., swd. That is, swd is strictly closer to C than is wd. By induction, S generates a subgroup of W transitive on chambers in a. Earlier we had shown that W is in bijection with chambers in a by the map w wc, so S must generate all of W. With fixed chamber C in an apartment a, the length l(w) of an element w W can be defined in two ways, that are not immediately identical: gallery length C to wc word length of w with respect to S where the word length of w is, by definition, the smallest n such that w = s 1 s 2... s n (with s i S) The word length certainly depends on choice of set S of generators. Proposition: Gallery length and word length are identical functions on W. In particular, for a shortest expression w = s 1... s n for w in terms of s i in S, the sequence of chambers is a minimal gallery from C to wc. C, s 1 C, s 1 s 2 C, s 1 s 2 s 3,..., s 1... s n C Proof: First, we check that these chambers are successively adjacent. Indeed, each pair of chambers is the image under s 1... s i of the pair s 1... s i C, s 1... s i s i+1 C C, s i+1 C The chamber s i+1 C is adjacent to C, and s 1... s i preserves adjacency, so the images are indeed adjacent. This shows that gallery length is less than or equal to word length. We prove the opposite inclusion by induction on minimum gallery length. Let C, C 1, C 2,..., C n = wc 15

16 Paul Garrett: Buildings (July 2, 2005) be a minimal gallery from C to wc. There is some s S such that C 1 = sc. Apply s to the gallery gives a gallery sc, C, sc 2,.... sc n = swc Thus, swc is strictly closer to C in gallery distance than was wc. That is, 1 + gallery length sw gallery length w Thus, by induction, the gallery length of sw is equal to the word length of sw. Visibly word length w = word length s sw 1 + word length sw Thus, since we already know that gallery length is at most word length, gallery length w word length w 1 + word length sw = 1 + gallery length sw gallery length w This proves the desired equality. Remark: A critical point missing from this little discussion of generation of W by reflections S is clarification of length of sw versus length of w, for w W and s S. For example, as it stands at the moment, it is hard to see why these lengths might not be the same. In fact, as we will see in the next section, things are as nice as we could hope: length sw = length w ± 1 This fact is critical for Hecke algebras and the Borel-Matsumoto theorem, and is non-trivial to prove. The technical discussion of foldings in the next section seems to be necessary to address this. 6. Reflections, foldings Still X is a thick building, but for the moment we can forget about groups acting upon it. The goal is to understand automorphism groups of apartments, without reference to any group that may be acting on the larger building. In particular, given two adjacent chambers in a building, we want a reflection automorphism which, by definition, should interchange the two adjacent chambers while fixing pointwise their common face. The existence of such a reflection on the subcomplex consisting just of the two chambers is immediate, but it is not at all obvious that this map extends to an automorphism of the apartment. A further technical surprise is that the construction (or proof of existence) of reflections (following Jacques Tits) uses foldings, defined and discussed below. The foldings themselves are constructed via the retractions of the whole building to various apartments. It is at this point that the thickness of the building itself is used to prove things about the apartments. As a by-product, we construct (many) retractions of an apartment to a given chamber within the apartment, thus proving that an apartment is label-able. Combining this with the (canonical) retraction of the building to an apartment, we prove that there always exist labelings. A folding of an apartment a in X is a chamber-complex map f : a a which is a retraction to its image and is two-to-one on chambers. [ ] The opposite folding f (if it exists) to a given folding f reverses the roles in each pair C, C of chambers that have the same image under f. That is, for f(c) = f(c ) = C, the opposite folding is f (C) = f (C ) = C. A folding with an opposite is reversible. Theorem: (Existence)Given two adjacent chambers C, D in an apartment a in a thick building X, there are (mutually opposite) foldings f : a a and g : a a such that f(c) = C = f(d) and g(c) = D = g(d). [ ] That is, the inverse image of any chamber consists of two chambers. 16

17 Letting H = f(a) and H = g(a), the restriction to H of f is an isomorphism f : H H. Similarly, the restriction to H of g is an isomorphism g : H H. We have a = H H, and on the other hand H H contains no chamber. Proof: Invoking the thickness, let B be a third chamber with face C D. Invoking the building axioms, let b be an apartment containing C and B. Let r b,c be the canonical retraction of X to b centered at C, and let r a,d be the canonical retraction of X to a centered at D. We claim that is the desired folding. f = r a,d r b,c E E E fig 9 D C img D D C D C building r b,c to apartment b r a,d to half-space First, r b,c (C) = C. And r a,d is a retraction to a, so is the identity on a, so maps C to itself. On the other hand, r b,c must map D to a chamber in b sharing the face C D (but not C), which must be B, by the thin-ness of b. Then r a,b maps r b,c (D) = B to a chamber in a sharing the face D B = D C (other than D), which must be C (by thin-ness). Thus, f(c) = C = f(d). Next, claim that f is an isomorphism on a minimal gallery C 0,..., C n = E from any chamber C 0 with face F = C D to a chamber E in X. To prove this, we prove that the two canonical retractions r b,c and r a,d have this property. Let r = r b,c. For E in b, r(c 0 ),..., r(e) = E is a gallery wholly within b, so the gallery distance in X from F to E a is equal to the gallery distance in b. [ ] Denote gallery distance in X by d X (, ) and gallery distance in an apartment b by d b (, ). We know that the restriction r : b b of the retraction r is an isomorphism (from the uniqueness lemma). For arbitrary E in X, let b be an apartment containing both C and E. We have d X (F, re) = d b (F, re) = d b (F, E) = d X (F, E) This verifies that r preserves gallery distances from F. The argument for r a,d is the same. Since F = C D is the common face, the composite f preserves gallery distances from F. Thus, the alleged folding f : a a is an isomorphism on minimal galleries in a from F = C D, since of course it cannot cause a minimal gallery to stutter without shortening it. For a minimal gallery C 0,..., C n = E in a from F (a face of C 0 ) to E (in a), C 0 is either C or D. For C 0 = C we have d X (F, E) = d X (C, E). Since f does not shorten this gallery, by the uniqueness lemma, f(e) = E (pointwise). Motivated by this, let H be the subcomplex of a consisting of all chambers E (and their faces) such that d X (F, E) = d X (C, E) Then f is the identity on H. On the other hand, for C 0 = D, application of f gives f(d) = C. That is, the image of the gallery begins with C, and is still minimal. That is, f(e) H, so f is indeed a retraction to H. [ ] Of course, a priori the gallery distance in b is greater than or equal to that in X. 17

18 Reversing the roles of C, D gives an analogous g : a a which is a retraction to the subcomplex H consisting of all chambers E (and their faces) such that d X (F, E) = d X (D, E) Since C and D are the only two chambers in a with face F, we are assured that a = H H but the nature of the intersection H H is not yet clear. Now show that H and H have no chamber in common. For E in Y Z, both f and g fix E pointwise. Let γ be a minimal gallery from F (a face of C 0 ) to E, in a. Then the images f(γ) and g(γ) are galleries from F to E. Since γ was already minimal, these galleries cannot stutter. By the uniqueness lemma, f = g, which is false, since f(c) = C D = g(c). Thus, there are no chambers in common. Last, we show that f : H H is an isomorphism, and g : H H is an isomorphism. This will also prove the two-to-one property of f and g. The map f g : H H maps C to itself pointwise. Galleries from F in H necessarily begin at C. Certainly (f g)(c) = C. Let γ be a minimal gallery from F to a chamber E. The composite f g does not shorten this gallery, so does not make it stutter. Thus, by the uniqueness lemma applied to f g : γ Y the map f g is an isomorphism on γ. In particular, the map f g fixes every E in H pointwise, so is the identity on H. Similarly, g f is the identity on H. Similarly, g f is the identity on H. The image of a folding is a half-space or half-apartment. Given adjacent chambers C, D in an apartment a, the folding f of a such that fc = C = fd is a folding along the codimension-one face C D of C. Several useful structural facts follow from the existence of reversible foldings. Proposition: Let f be a reversible folding of a, with opposite folding g. Let C, D be adjacent chambers such that f(c) = C = f(d). Then the half-space H = f(a) is H = {chambers E a : d(c, E) < d(d, E)} where d(, ) denotes the length of shortest gallery connecting two chambers. Proof: First, given a chamber E in a with fe E, we claim that the image fc,..., fe of any gallery C = C 0,..., C n = E in a connecting C and E stutters. Certainly there is an index i such that fc i = C i and fc i+1 C i+1. Since a is thin, fc i+1 has no choice but to be C i. That is, the image of the gallery stutters. In particular, fe is strictly closer to fc = C than is E (in minimum-gallery distance). Similarly, for E H, and γ a minimal gallery from D to E, fγ is a stuttering gallery from fd = C to fe = E, so can be shortened. Thus, d(c, E) < d(d, E). The other half of the assertion follows from the presence of the opposite folding. Corollary: Every thick building has a labeling. [ ] Proof: It suffices to prove that a building has a retraction to a given chamber C, since this gives a labeling. We already know that there is a (canonical) retraction of the whole building to an apartment, so it suffices [ ] Since every thick building has a labeling, we would not need to explicitly assume label-ability. However, in practice, we ll usually have an obvious labeling arising from external circumstances. 18

19 to exhibit a retraction of an apartment a to a given chamber within it. This last part is non-canonical. Let f 0, f 1,..., f n be the foldings along the codimension-one faces of C, and let ϕ = f 0 f 1... f n Let D i be the chamber adjacent to C such that f i is a folding along C D i. From the previous proposition, any chamber closer to D i than to C is moved closer by f i, while chambers closer to C than to D i are not moved at all by f i. A given chamber E in a has a minimal gallery from C, and some D i must be the second gallery in this chamber, so the corresponding f i moves E strictly closer to C, and moves no chamber in a farther from C. Thus, the composition ϕ of all the foldings along the codimension-one faces of C is a simplicial complex map which moves every chamber in a strictly closer to C. Thus, chambers at gallery distance l from C will be mapped to C by the l-fold composite ϕ l. The function on vertices defined as ϕ (x) = ϕ l (x) (for l sufficiently large) where sufficiently large means such that ϕ l (x) is in C is well-defined, since ϕ fixes C pointwise. This gives the retraction of a to C. Corollary: (Uniqueness) Given adjacent chambers C, D in an apartment a, there is a unique reversible folding f : a a such that fc = C = fd. Proof: Existence is the content of the theorem above. The proposition characterizes the half-space H = f(a) intrinsically. Similarly, for g the opposite folding to f, the opposite half-space H = g(a) is similarly characterized. On H, the folding f is the identity. On H the folding f is determined completely on D, and is an isomorphism on H, so is completely determined, by the uniqueness lemma. Thus, f is unique. At last we can construct reflections, in terms of foldings, independently of any assumption of a group action on the building. Corollary: Given adjacent chambers C, D in an apartment a in a thick building X, there is a unique automorphism s : a a such that s fixes C D pointwise, sc = D, and sd = C. This s is the reflection of a along C D. It follows that s 2 = 1. This reflection is given explicitly as follows. Let f be the folding with fc = C = fd and g its opposite, and half-spaces f(a) = H and g(a) = H. Then for vertices x in a. s(x) = { fx (for x H ) gx (for x H) Proof: Any automorphism s of a sends non-stuttering galleries to non-stuttering galleries. For s fixing C D pointwise and interchanging C and D, s is determined on C, so is completely determined, from the uniqueness lemma. Since s 2 C = C and s 2 fixes the codimension-one face C D of C, it must be that s 2 fixes C pointwise, so s 2 = 1 on a, again by uniqueness. For existence, it remains to show that the indicated formula satisfies the conditions. First, the theorem showed that f : H H is an isomorphism, and that g : H H is an isomorphism. And f is a retraction to H, so is the identity on H, and similarly for g on H. Thus, on H H both f and g are the identity, which allows us to piece together s as in the indicated formula. [ ] [ ] Implicit in this piecing-together is the fact that H H, H, and H are simplicial subcomplexes of the apartment, and f, g are simplicial complex maps. 19